\(\chi \) )= (2, 5, ) =   S, T S2 = (ST)5 = 1   .

It is the first in family of Teichmüller curves discovered by Veech in [1]}. The papers [2]}, [3]}, and [4]} give an algebraic description of \(\phi \) , which we recall.
Let \(\widetilde{\mathchoice{\chi }{}{}{}}{\chi }\) := 1 - 5\(, where \) 5\( is the set of fifth roots of unity. Let \) DAut(\(\chi \)\(\chi \) )\( be the subgroup generated by the maps \) t1t\( and \) t 2 t\(, where \) t\( is the restriction of the global rational coordinate on \) 1\( to \)\(\chi \)\(\chi \)\(, and \) :=(2 5 -1)\(. For any \) t \(\chi \)\(\chi \)\(, let \) Ft\( be the unique smooth Riemann surface of genus two associated with the polynomial\begin{displaymath}y^2=\prod _{i=1}^5(x - \zeta ^i - \zeta ^{-i}t) = x^5 - 5tx^3 + 5t^2x - t^5 - 1.\end{displaymath}It turns out that \) Ft\( is isomorphic to \) Fgt\( for all \) g D\(, and that \)\(\chi \)\(\chi \)\(\chi \)\(\chi \) /D\(. Under this identification, the map \)\(\chi \)\(\chi \) M2\( is simply given by \) ([t])=[Ft]\(, for \) [t] \(\chi \)\(\chi \) /D\(.\)
